(a−−√+8a−−√)2 a + 8 a ) 2 = 54 + b2–√ 2 , a and b are positive integers. Find the value of a and the value of b.

To solve for the values of a and b, given the equation

(√a + √(8a))² = 54 + 2√(b²),

First, let's simplify the equation:

1. Expand the left-hand side:

(√a + √(8a))² = a + 2√a√(8a) + 8a = a + 2√(8a²) + 8a = a + 8√a² + 8a = 9a + 8a.

This results in: 9a + 8√a² = 54 + 2√(b²)

2. Simplify the right-hand side:

The right side can be further simplified as 54 + 2b (since 2√(b²) equals 2b).

Now, the equation is: 9a + 8√a² = 54 + 2b

Since a and b are integers, we attempt integer values that satisfy both sides of the equation. From the possible options:

If a = 6:
9 * 6 + 8 * √6² = 54 + 12
54 + 48 = 54 + 12b
102 = 54 + 12b

This does not satisfy. Let's try another pair.

If a = 24:
9 * 24 + 8 * √24² = 54 + 2b
216 + 192 = 54 + 2b
408 = 54 + 2b
2b = 408 - 54
2b = 354
b = 177

This would be very large for b. Let's try another option.

If a = 2:
9 * 2 + 8 * √2² = 54 + 2b
18 + 16 = 54 + 2b
34 ≠ 54 + 2b

If a = 6:
9 * 6 + 8 * √6² = 54 + 2b
54 + 48 = 54 + 12b
102 = 54 + 2b
2b = 102 - 54
2b = 48
b = 24

Thus, the correct values are a = 6 and b = 24.

This matches the given option set for a = 6 and b = 24.